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Dynamical behavior analysis and bifurcation mechanism of a new 3-D nonlinear periodic switching system

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Abstract

This paper presents a new periodic switching chaotic system, which is topologically non-equivalent to the original sole chaotic systems. Of particular interest is that the periodic switching chaotic system can generate stable solution in a very wide parameter domain and has rich dynamic phenomena. The existence of a stable limit cycle with a suitable choice of the parameters is investigated. The complex dynamical evolutions of the switching system composed of the Rössler system and the Chua’s circuit are discussed, which is switched by equal period. Then the possible bifurcation behaviors of the system at the switching boundary are obtained. The mechanism of the different behaviors of the system is investigated. It is pointed out that the trajectories of the system have obvious switching points, which are decided by the periodic signal. Meanwhile, the system may be led to chaos via a period-doubling bifurcation, resulting in the switching collisions between the trajectories and the non-smooth boundary points. The complicated dynamics are studied by virtue of theoretical analysis and numerical simulation. Furthermore, the control methods of this periodic switching system are discussed. The results we have obtained clearly show that the nonlinear switching system includes different waveforms and frequencies and it deserves more detailed research.

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References

  1. Xu, J., Chargé, P., Fournier-Prunaret, D., Taha, A.K., Long, K.P.: Chaos generator for secure transmission using a sine map and an RLC series circuit. Sci. China, Ser. F, Inf. Sci. 53, 129–136 (2010)

    Google Scholar 

  2. Putyrski, M., Schultz, C.: Switching heterotrimeric G protein subunits with a chemical dimerizer. Chem. Biol. 18(9), 1126–1133 (2011)

    Article  Google Scholar 

  3. Kim, S., Kim, Y.-C., Yoon, B.-Y., Kang, M.: An integrated congestion control mechanism for optimized performance using two-step rate controller in optical burst switching networks. Comput. Netw. 51(3), 606–620 (2007)

    Article  MATH  Google Scholar 

  4. Branicky, M.S.: Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43, 475–482 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Xie, G., Wang, L.: Stabilization of switched linear systems with time-delay in detection of switching signal. J. Math. Anal. Appl. 305(1), 277–290 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Sun, Z., Zheng, D.Z.: On reachability and stabilization of switched linear systems. IEEE Trans. Autom. Control 46, 291–295 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, D., Guo, L., Lin, Y., Wang, Y.: Stabilization of switched linear systems. IEEE Trans. Autom. Control 50(5), 661–666 (2005)

    Article  MathSciNet  Google Scholar 

  8. Huang, H., Qu, Y., Li, H.-X.: Robust stability analysis of switched Hopfield neural networks with time-varying delay under uncertainty. Phys. Lett. A 34(12), 345–354 (2005)

    Article  MathSciNet  Google Scholar 

  9. Pettersson, S., Lennartson, B.: Stability and robustness for hybrid systems. In: Proc. of the 35th IEEE Conference on Control and Decision, Japan, pp. 1202–1207 (1996)

    Chapter  Google Scholar 

  10. Leine, R.I.: Bifurcations of equilibria in non-smooth continuous systems. Physica D 223, 121–137 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bhattacharyya, R., Mukhopadhyay, B.: On an eco-epidemiological model with prey harvesting and predator switching: local and global perspectives. Nonlinear Anal., Real World Appl. 11, 3824–3833 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, L.-l., Feng, J., Dimirovski, G.M., Zhao, J.: Reliable stabilization and H∞ control for switched systems with faulty actuators: an average dwell time approach. In: 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA, 10–12 June 2009

    Google Scholar 

  13. Stikel, G., Bokar, J., Szabo, Z.: Necessary and sufficient condition for the controllability of switching linear hybrid systems. Automatica 40(6), 1093–1097 (2004)

    Article  MathSciNet  Google Scholar 

  14. Xiang, Z., Sun, Y.-n., Chen, Q.: Robust reliable stabilization of uncertain switched neutral systems with delayed switching. Appl. Math. Comput. 21, 79835–79844 (2011)

    MathSciNet  Google Scholar 

  15. Sharan, R., Banerjee, S.: Character of the map for switched dynamical systems for observations on the switching manifold. Phys. Lett. A 372(23), 4234–4240 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Stikel, G., Bokar, J., Szabo, Z.: Necessary and sufficient condition for the controllability of switching linear hybrid systems. Automatica 40(6), 1093–1097 (2004)

    Article  MathSciNet  Google Scholar 

  17. Mancusia, E., Altimarib, P., Russoc, L., Crescitelli, S.: Multiplicities of temperature wave trains in periodically forced networks of catalytic reactors for reversible exothermic reactions. Chem. Eng. J. 171(10), 655–668 (2011)

    Article  Google Scholar 

  18. Olejnik, P., Awrejcewicz, J.: Application of Hénon method in numerical estimation of the stick–slip transitions existing in Filippov-type discontinuous dynamical systems with dry friction. Nonlinear Dyn. (2013). doi:10.1007/s11071-013-0826-7

    MATH  Google Scholar 

  19. Hiskens, I.A.: Stability of hybrid system limit cycles: application to the compass gait biped robot. In: Proc. of the 40th IEEE Conference on Decision and Control, Orlando, pp. 774–779 (2001)

    Google Scholar 

  20. Gamez-Guzman, L., Cruz-Hernández, C., López-Gutiérrez, R.M., García-Guerrero, E.E.: Synchronization of Chua’s circuits with multi-scroll attractors. Commun. Nonlinear Sci. Numer. Simul. 21(9), 2765–2775 (2009)

    Article  Google Scholar 

  21. Kousaka, T., Ueta, T., Ma, Y., Kawakami, H.: Control of chaos in a piecewise smooth nonlinear system. Chaos Solitons Fractals 36(7), 1019 (2006)

    Article  Google Scholar 

  22. Yu, Y., Zhang, C., Han, X.J., Bi, Q.S.: Oscillations and their mechanism of compound system with periodic switches between two subsystems. Acta Phys. Sin. 20, 200501–200507 (2012)

    Google Scholar 

  23. Han, X.J., Jiang, B., Bi, Q.S.: 3-torus, quasi-periodic bursting, symmetric subHopf/fold-cycle bursting, subHopf/fold-cycle bursting and their relation. Nonlinear Dyn. 61, 667–676 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, T., Yang, C.-M., Yang, L.-B.: Control of Rössler system to periodic motions using impulsive control methods. Phys. Lett. A 232(5), 356–361 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gonzalo, M., Ramírez-Ávila, J.A., Gallas, C.: How similar is the performance of the cubic and the piecewise-linear circuits of Chua. Phys. Lett. A 375(2), 143 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are supported by the Natural Science Foundation of China (Grant Nos. 21276115 and 11202085) and the Research Foundation for Advanced Talents of Jiangsu University (Grant Nos. 11JDG065 and 11JDG075).

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Authors and Affiliations

  1. Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, Jiangsu, 212013, P.R. China

    Yue Yu, Chun Zhang, Xiujing Han & Qinsheng Bi

  2. Faculty of Science, Nantong University, Nantong, Jiangsu, 226019, P.R. China

    Yue Yu

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  1. Yue Yu
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  2. Chun Zhang
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  4. Qinsheng Bi
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Correspondence to Qinsheng Bi.

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Yu, Y., Zhang, C., Han, X. et al. Dynamical behavior analysis and bifurcation mechanism of a new 3-D nonlinear periodic switching system. Nonlinear Dyn 73, 1873–1881 (2013). https://doi.org/10.1007/s11071-013-0910-z

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